Molecules in solution are generally characterized by their weight averaged molar mass, their mean square radius, and the second virial coefficient; the latter being a measure of the interaction between the molecules and the solvent. For unfractionated solutions, these properties may be determined from measurement of the manner by which they scatter light following the method described by Bruno Zimm in his seminal 1948 paper which appeared in the Journal of Chemical Physics, volume 16, pages 1093 through 1099. Basically, the light scattered from a small volume of the solution is measured over a range of angles and concentrations. The properties derived from the light scattering measurements are related through the formula developed by Zimm, as follows:
R(xcex8)/K*=MWcP(xcex8)[1xe2x88x922A2MWcP(xcex8)]+O(c3)+L,xe2x80x83xe2x80x83(1)
where MW is the weight average molar mass, R(xcex8) is the measured excess Rayleigh ratio, P(xcex8) is the form factor of the scattering molecules, K*=4xcfx802(dn/dc)2n02/(Naxcex04), Na is Avogadro""s number, (dn/dc) is the refractive index increment, n0 is the solvent refractive index, and xcex0 is the wavelength of the incident light in vacuum. The collection of light scattering data over a range of scattering angles is referred to more commonly as multiangle light scattering, or by the acronym MALS. The data so-collected are then extrapolated to vanishingly small concentrations and 0xc2x0 scattering angles by means of the so-called Zimm plot technique. For this purpose, the reciprocal of Eq. (1) is more commonly used which, through order c2, may be written as follows:
K*c/R(xcex8)=1/[MWP(xcex8)]+2A2c.xe2x80x83xe2x80x83(2)
The Zimm plot technique was developed primarily for binary solutions comprised of a simple solvent and the molecular solute. To apply the technique to more complex solvents containing buffering salts, for example those used for the study of proteins by light scattering, the solutions must be dialyzed at each measured concentration. For these latter solutions, the procedure is both labor and time intensive.
A more powerful means by which a molecular solution may be analyzed is to fractionate the sample first by chromatographic means, such as size exclusion chromatography or SEC, and then perform a Zimm plot on each eluting fraction or slice. Because such SEC separations are subject to very large dilutions, the sample concentration is so small at the time the light scattering measurement is performed that the need to extrapolate to vanishingly small concentrations is obviated since the concentration is already almost negligible. The only extrapolation required is that to zero scattering angle which is easily performed by software such as the ASTRA(copyright) software developed by Wyatt Technology Corporation of Santa Barbara, Calif. However, this approach is tantamount to assuming A2=0, which essentially precludes its determination.
There are several advantages to this fractionation approach in addition to the obvious simplifications of the Zimm technique. First is the ability of the combined fractionation/MALS measurement to permit calculation of the distributions of molar mass and mean square radius over the entire sample. From these distributions, their associated moments, such as the weight averaged, number averaged, and z-averaged molar mass and sizes, may be calculated. Details of the chromatographic separation methods, the definitions and calculations of the mass and size moments, and an explanation of the terminology used to describe the associated distributions may be found in the 1993 review article by Wyatt in Analytica Chimica Acta, volume 272, pages 1 through 40. It should be noted that the weight average molar mass of the sample calculated from the fractionated sample measurements should be nearly identical to the corresponding weight average molar mass generated from batch measurements performed by the Zimm plot technique. A small discrepancy between the two methods is due to the setting of A2=0 in the chromatographic approach. Another benefit of such measurement is that the sample undergoing SEC fractionation is being dialyzed throughout its separation permitting, thereby, MALS measurement with buffered solutions. Unfortunately, since the molecular solute was assumed to be at a vanishingly small concentration, in general there has been no means to recover the second virial coefficient for the solvent/solute interaction. Indeed, until the development of the present invention, the only means by which the 2nd virial coefficient could be derived was from the analysis of unfractionated samples following the Zimm plot technique.
In U.S. Pat. No. 5,129,723 by Howie, Jackson, and Wyatt, a method was described whereby an unfractionated sample was injected into a MALS detector following dilution and thorough mixing. This procedure produced a sample peak passing through the light scattering detector whose profile was assumed proportional to the concentration profile of the diluted, yet unfractionated, sample. Since the mass distribution at each slice was the same, it was assumed that each point of the profile was proportional, at that point, to the sample""s concentration times the weight averaged molar mass by referring to Eq. (2) and setting A2=0. On this basis, a Zimm plot could be produced using a set of these points and the associated weight average molar mass, mean square radius, and 2nd virial coefficient were then derived. A concentration detector was not needed, since knowledge of the total mass injected was sufficient to convert the sample peak curve into a concentration profile. The method was flawed because the assumption that A2 was zero contradicted the derived result that it was not. Selecting concentration points from the ascending or descending parts of the peak yielded different results while using concentration points from both sides produced extremely poor and inconsistent Zimm plots.
Returning to the fractionation/dialysis approach for the case of a protein sample, we note that the weight average molar mass at each eluting fraction should be constant throughout the elution peak since, absent aggregation, the protein mass distribution is monodisperse. Even in the presence of aggregates, the separation method should be developed to separate such aggregates from the protein monomer. With the addition of a concentration monitor, such as a differential refractive index detector, an evaporative light scattering detector, or UV detector, once the delay volume between the light scattering and concentration detectors has been established, both the light scattering and concentration signals will be known at each eluting fraction or slice. The delay volume may be determined, for example, by the method described by Wyatt and Papazian in their 1993 paper appearing in volume 11, pages 862 through 872 of the trade journal LC-GC. From the thus-corrected MALS and concentration data, a Zimm plot might be generated from values at several different slices or sets of slices of the elution profile. On this basis, the weight average molar mass and 2nd virial coefficient may be derived therefrom. In general, the root mean square radius for most proteins will be too small to be derived from said Zimm plot.
There remains a relatively small, though important, distortion associated with Zimm plots derived in this manner: because of band broadening effects, the actual concentration profile will appear slightly flattened, i. e. spread out. The term xe2x80x9cband broadeningxe2x80x9d refers to the observed broadening of a peak""s breadth due to the presence of the additional dead volumes between detectors. Part of such a broadening effect arises from the delay volume itself, though this is usually small compared to the larger dead volume, for example, of the DRI detector needed to assure thermal equilibrium of the samples passing therethrough. Because of the slight distortions of the resultant concentration profile, the derived mass distribution will no longer be calculated as monodisperse despite the monodispersity of the protein sample. The other properties derived, such as the 2nd virial coefficient, will contain also some deviations from their true values because of this effect. Since the 2nd virial coefficient is generally very small, it produces a relatively small contribution to the final mass calculation. But the error associated with band broadening upon the mass calculation is a comparably small quantity and so one would expect that such band broadening could affect the 2nd virial coefficient calculation significantly. Various analytical corrections of such band broadening have been developed over the years, but they are not without their own problems. Even were such corrections made suitably, the need to make a Zimm plot solely for the purpose of measuring the 2nd virial coefficient of a particular protein-solvent interaction is time consuming. Since the protein molar mass for a monodisperse sample in a suitable buffering solvent is easily measured by MALS, mass spectroscopy, or direct sequencing, it would be useful to circumvent the Zimm plot altogether and measure the 2nd virial coefficient directly. This would speed up the measurement process considerably and permit the examination of many solvent/protein combinations to derive conditions for protein crystallization or solubility. Such determinations are essential to provide a better understanding of protein processes. A particular beneficiary of such rapid measurements would be the field of combinatorial chemistry. It is a major objective of this invention to show a new means by which such determinations may be achieved.
This invention concerns a method by which the 2nd virial coefficient of a sample comprised of a monodisperse molar mass distribution in a solvent may be determined directly. The same method may be applied to certain classes of unfractionated samples. It is of particular importance in the field of protein chemistry wherein light scattering techniques are applied to measure molar mass interactions with complex buffered solvents and require that samples be dialyzed against such solvents prior to such measurement.
The method, applied to the exemplar of a monodisperse protein sample, begins with the preparation of the sample for injection onto a SEC column set. Following the columns, a MALS detector and concentration detector are connected serially. For small molecules whose mean square radii are too small to measure, measurement at a single scattering angle, such as 90xc2x0 may suffice, though the precision of the determination may be diminished. These are the conventional elements of a standard separation by SEC means resulting in an absolute determination of the eluting molar masses present in the sample. Unlike this conventional measurement procedure, the direct determination of the 2nd virial does not require correction for the delay volume between detectors nor is it affected by the presence of band broadening.
From the measurement of the concentration, c1, at each slice i, the square of this quantity, ci2, is calculated. The total eluted mass, m, is then calculated by integrating the area under the concentration detector response peak as a function of elution volume. The excess Rayleigh ratios for each scattering angle are extrapolated to zero scattering angle and the sums             ∑      i        ⁢                            R          i                ⁢                  (                      0            ⁢            xc2x0                    )                    ⁢      Δ      ⁢              xe2x80x83            ⁢                        v          i                /                  K          *                      ,      xe2x80x83    ⁢            ∑      j        ⁢                  c        j            ⁢              xe2x80x83            ⁢      Δ      ⁢              xe2x80x83            ⁢              v        j              ,
and       ∑    j    ⁢            c      j      2        ⁢          xe2x80x83        ⁢    Δ    ⁢          xe2x80x83        ⁢          v      j      
are calculated, where xcex94xcexdi is the incremental elution volume of slice i. Note that the contributions to the summations over cj and cj2 may include a greater number of slice contributions than the sums over the excess Rayleigh ratios since the concentration peak is often broadened due the presence of band broadening. For the case of equidistant slices, i. e. when xcex94xcexdi=xcex94xcexd=constant, the sums calculated are simplified to             ∑      i        ⁢                            R          i                ⁢                  (                      0            ⁢            xc2x0                    )                    /              K        *              ,      xe2x80x83    ⁢            ∑      j        ⁢                  c        j            ⁢              xe2x80x83            ⁢      Δ      ⁢              xe2x80x83            ⁢              v        j              ,
and       ∑    j    ⁢            c      j      2        ⁢          xe2x80x83        ⁢    Δ    ⁢          xe2x80x83        ⁢                  v        j            .      
From these measurements and calculations, the 2nd virial coefficient may be derived immediately.